Extended affine Lie algebras
Extended affine Lie algebras
复制标题
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
E. Neher
中科院分区：
文献类型：
--
作者：
E. Neher
在这个公告中，我们描述了一个扩展的仿射李代数的结构，其无中心的核心。0.导论.广义仿射李代数是一类复李代数，包括有限维单李代数、仿射李代数和环面李代数。它们与Saito的椭圆李代数（[29]）密切相关。最初由物理学家Høegh-Krohn和B提出。Torrésani [20]在不可约拟单李代数的名称下，扩展仿射李代数已经在Allison，Azam，Berman，Gao和Pianzola的AMS-回忆录[2]中建立了坚实的数学基础。特别是，人们可以找到有一个详细的研究根系出现在扩展仿射李代数。这些李代数的各种类的结构和表示理论已经在许多论文中进行了研究，见第4节（可能不完整）的调查。本文主要讨论广义仿射李代数的结构。请读者参阅本说明的正文以了解确切的定义，我们在本导言中只对有关结构作一个粗略的概述。扩张仿射李代数的两个重要性质是存在一个不变的非退化形式和一个有限维自中心可对角化子代数H。因此E有一个根空间分解E =<$E <$$>和一个根系R，由那些<$∈ H <$且E <$6 = 0组成。E上的形式给出了将R = R <$R划分为各向同性根R和各向异性根R，在仿射情况下将分解推广为虚根和真实的根。设Ec是{E ∈ Ran：E ∈ Ran}生成的理想，称为E的核.假设E可以从其核Ec恢复，在这个意义上，自然表示E → DerEc：x 7→ adx的核|Ec躺在Ec里。核心Ec可能有一个非平凡的中心，并且证明更容易描述其中心商L = Ec/Z（Ec），其中Z（Ec）表示Ec的中心。因此，情况可以用下图Ec E来概括
In this announcement we describe the structure of an extended affine Lie algebra in terms of its centreless core. 0. Introduction. Extended affine Lie algebras are a class of complex Lie algebras that includes finite-dimensional simple Lie algebras, affine Lie algebras and toroidal Lie algebras. They are closely related to Saito’s elliptic Lie algebras ([29]). Originally proposed by the physicists Høegh-Krohn and B. Torrésani [20] under the name irreducible quasi-simple Lie algebras, extended affine Lie algebras have been put on a sound mathematical footing in the AMS-memoirs [2] by Allison, Azam, Berman, Gao and Pianzola. In particular, one can find there a detailed study of the root systems appearing in extended affine Lie algebras. The structure and representation theory of various classes of these Lie algebras has since been investigated in many papers, see section 4 for a (probably incomplete) survey. In this note we will describe the structure of extended affine Lie algebras in general. Referring the reader to the main body of this note for precise definitions, we will only give a rough sketch of the relevant structures in this introduction. Two important properties of an extended affine Lie algebra are the existence of an invariant nondegenerate form and a finite-dimensional selfcentralizing ad-diagonalizable subalgebra H. Thus E has a root space decomposition E = ⊕Eξ and a root system R, consisting of those ξ ∈ H∗ with Eξ 6= 0. The form on E gives rise to a partition R = R ∪ R into isotropic roots R and anisotropic roots R, generalizing the decomposition into imaginary and real roots in the affine case. Let Ec be the ideal generated by {Eξ : ξ ∈ Ran}, called the core of E. One assumes that E can be recovered from its core Ec in the sense that the kernel of the natural representation E → DerEc : x 7→ adx|Ec lies in Ec. The core Ec may have a non-trivial centre, and it turns out to be easier to describe its central quotient L = Ec/Z(Ec), where Z(Ec) denotes the centre of Ec. The situation can thus be summarized by the following diagram Ec E